numerical investigation of sooting propensity of ethanol ... · numerical investigation of sooting...

Numerical Investigation of Sooting Propensity ofEthanol at Elevated Pressures

by

Vishal Singh

A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science

Graduate Department of Aerospace EngineeringUniversity of Toronto

Copyright © 2014 by Vishal Singh

Abstract

Numerical Investigation of Sooting Propensity of Ethanol atElevated Pressures

Vishal Singh

Masters of Applied Science

Graduate Department of Aerospace Engineering

University of Toronto

2014

The effects of pressure on the combustion properties of ethanol were investigated. One-dimensional

analysis was employed to evaluate the fundamental combustion properties of ethanol as a func-

tion of equivalence ratio and pressure. Two mechanisms were used in the one-dimensional

simulation. The predicted results displayed good agreement with experimental data. Further,

a state of the art computational framework was used to simulate the two-dimensional, ax-

isymmetric, co-flow laminar diffusion flames of ethanol. The framework is a highly-scalable

combustion modelling tool designed specifically for use on large multi-processor parallel com-

puter systems. The flame structure predictions were obtained for ethanol. The soot volume

fraction had an annular structure and maximum concentrations occured in the annular region.

In agreement with the past theoretical and numerical measurements for other gaseous fuels,

the current work showed a constant flame height and an increasing trend of soot yield with

increasing pressure.

ii

Acknowledgements

I would like to sincerely thank Professor C.P.T. Groth for giving me the chance to study at the

University of Toronto’s Insitute for Aerospace Studies under his direction. His guidance and

support through the course of my studies has been greatly appreciated.

Furthermore, I would like thank Dr. Marc Charest in helping me understand the code. I would

also like to thank Dr. Scott Northrup in helping me with the conjugate heat transfer develope-

ment. I have learned a great deal in object-oriented programming and parallel programming

from him.

I would also like to thank my friends at UTIAS CFD lab who have made my experience here

truly enriching. I particular I would like to thank Adam, Martin, Sandipan and Shahriar.

Lastly, I would like to thank my parents, sister and brother for their support and prayers.

Toronto, 2014 Vishal Singh

iii

Contents

Abstract ii

Acknowledgements iii

Contents iv

List of Figures vi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Soot Formation, Oxidation and Modelling . . . . . . . . . . . . . . . . . 3

1.2.3 Effects of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Numerical Investigation of One-Dimensional Laminar Premixed Flames . 6

1.3.2 Numerical Simulation of Two-Dimensional Axisymmetric

Laminar Co-Flow Diffusion Flames . . . . . . . . . . . . . . . . . . . . . . 7

2 Numerical Solution Methods 10

2.1 One-Dimensional Laminar Premixed

Flame Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

iv

2.1.1 Cantera Solution Method for Flame Speed

and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Axisymmetric Laminar Co-Flow Diffusion

Flame Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Finite-Volume Solution Discretization Method . . . . . . . . . . . . . . . 14

2.2.3 Low-Mach-Number Preconditioning . . . . . . . . . . . . . . . . . . . . . 15

2.2.4 Round-Off Error Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.5 Inviscid Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.6 Higher-Order Spatial Accuracy for Inviscid Fluxes . . . . . . . . . . . . . 18

2.2.7 Viscous Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.8 Steady State Relaxation Method . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.9 Radiation Transfer Equation (RTE) . . . . . . . . . . . . . . . . . . . . . 20

2.2.10 Discrete Ordinates Method (DOM) . . . . . . . . . . . . . . . . . . . . . 22

2.2.11 Parallel Block-Based Solution Scheme for RTE . . . . . . . . . . . . . . . 23

2.2.12 Overall Solution Algorithm for RTE . . . . . . . . . . . . . . . . . . . . . 24

2.2.13 Soot Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.14 Soot Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Results I: One-Dimensional Laminar Premixed Flames 27

3.1 Benchmark Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Laminar Flame Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Effects of Chemical Kinetic Mechanism . . . . . . . . . . . . . . . . . . . 28

3.2.2 Effects of Stoichiometry and Pressure . . . . . . . . . . . . . . . . . . . . 34

3.3 Adiabatic Flame Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Results II: Laminar Co-Flow Diffusion Flames for Ethanol 39

v

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Discretization and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Solution Procedure for Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Predicted Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6.1 Contours of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6.2 Radial Temperature Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Prediction of Flame Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.8 Prediction of Flame Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.9 Concentrations of Intermediate Species . . . . . . . . . . . . . . . . . . . . . . . . 49

4.9.1 Acetylene Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.9.2 Carbon Monoxide Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . 49

4.10 Predicted Soot Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.10.1 Contours of Soot Volume Fraction . . . . . . . . . . . . . . . . . . . . . . 52

4.10.2 Radial Profiles of Soot Concentration . . . . . . . . . . . . . . . . . . . . 52

4.10.3 Sooting Propensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusions and Future Research 56

5.1 Conclusions I: One-Dimensional Laminar

Premixed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Conclusions II: Axisymmetric Laminar Co-Flow

Diffusion Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . . . 57

References 65

vi

List of Figures

1.1 UITAS high-pressure laminar co-flow diffusion flame burner apparatus. . . . . . . 7

2.1 Schematic diagram showing generic 2D quadrilateral computational cell. . . . . . 15

2.2 Schematic diagram showing diamond path viscous flux reconstruction stencil for

a quadrilateral computational cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Comparison of experimental data of laminar flame speed to predicted values from

Cantera for ethanol using the Dryer mechanism at 1 atm and 5 atm. . . . . . . . 28

3.2 Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values

from Cantera at 1 atm and 5 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . 30


from Cantera at 10 atm and 15 atm. . . . . . . . . . . . . . . . . . . . . . . . . . 31


from Cantera at 20 atm and 25 atm. . . . . . . . . . . . . . . . . . . . . . . . . . 32


from Cantera at 1 atm and 25 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Comparison of calculated values of laminar flame speed from Cantera for pres-

sures between 1 atm and 25 atm for Dryer and Reduced Dryer mechanism. . . . 35

3.7 Laminar flame speed as a function of pressure comparison with theory. . . . . . . 36

3.8 Adiabatic flame temperature of Dryer and Reduced Dryer mechanism as a func-

tion of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

4.1 University of Toronto Institute for Aerospace Studies (UTIAS) high-pressure

laminar co-flow diffusion burner schematic. . . . . . . . . . . . . . . . . . . . . . 40

4.2 Schematic of computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 2D computational grid showing the clustering of computational cells. The mesh

contains 61,080 cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Typical convergence history for ethanol-air flames. The normalized L2–norm of

the continuity equation is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Experiemental photographs of co-flow diffusion flames of approximately 5 mm in

height for ethanol at pressures of 1 atm and 5 atm as obtained by Karatas [14]. . 44

4.6 Effect of reaction mechanism and pressure on flame structure for ethanol at

pressures of 1, 5, 10 and 15 atm. Peak temperature is indicated on the bottom

right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 Radial temperature profiles at three axial locations for ethanol for pressures 5,

10 and 15 atms using Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Flame radius as a function of pressure for ethanol using Dryer mechanism. . . . . 49

4.9 Effect of reaction mechanism and pressure on acetylene mass fraction for ethanol

at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the

bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.10 Effect of reaction mechanism and pressure on carbon monoxide mass fraction for

ethanol at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated

on the bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.11 Effect of reaction mechanism and pressure on soot volume fraction for ethanol

at pressures of 5, 10 and 15 atmospheres. Peak soot volume fraction is indicated

on the bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.12 Radial soot volume fraction at three axial locations for ethanol for pressures of

5, 10 and 15 atm using Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . 54

4.13 Effect of pressure on the maximum carbon conversion factor for ethanol using

Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

viii

Chapter 1

Introduction

1.1 Motivation

The majority of practical combustion devices such as industrial furnaces, gas turbine combustors

and internal combustion engines burn high carbon-content fossil fuels. However, the current

sources of fossil fuels are rapidly declining [1]. In addition, there are serious concerns regarding

the emissions such as carbon dioxide, carbon monoxide, soot and other particulates from the

use of fossil fuels that are harmful to human health and the environment. As the global demand

for energy rises, there is an increasing concern for fuel security and environmental impacts that

result from the use of fossil fuels to meet the demand [2]. Thus, there is an urgent need to

identify renewable fuel sources to meet rising global demand. Biofuels are an attractive option

since they are renewable and are potentially more environmentally friendly [3].

For optimal efficiency and weight, combustion in practical gas turbines is turbulent and is

carried out at high pressures. Operating at elevated pressures is desired since the combustion

intensity (energy released per unit volume) is proportional to the square of pressure. However,

the pressure can have an impact on engine emissions. In particular, pressure has significant

effect on the overall soot production by directly affecting the production and oxidation rates

of soot in turbulent diffusion flames of typical aviation gas turbine engines. These flames are

difficult to characterize due to experimental limitations related to optical accessibilty, complex

flame geometries and different time and length scales involved. For these reasons, laminar

flames are often studied since they provide easily controlled conditions and the results can be

projected to practical turbulent flames using the flamelet approach [4].

Focusing on soot, these carbonaceous particles that are emitted in the upper atmosphere from

1

Chapter 1. Introduction 2

aircraft engines affect the global thermal balance by either absorbing sunlight or by contribut-

ing to contrail formation [5]. The presence of soot also adversely affects the performance of

propulsion systems. Soot radiation is the dominant mechanism for the spread of unwanted fires

and is a major heat load on combustor components causing reliability issues. Finally, soot itself

can be toxic to human life and soot particles are strongly associated with detrimental health

effects. In the Earth’s atmosphere, soot contributes to the entrapment of solar radiation that

is believed to lead to global warming [6].

While obviously a detrimental pollutant, there are very few fundamental studies on soot forma-

tion and there is no complete framework to charactize its behaviour under different conditions.

In recent years, there have been some studies that have been carried out under atmospheric

conditions but they do not represent accurately the conditions inside practical gas turbines.

Some recent experimental studies have been carried out at University of Toronto Institute for

Aerospace Studies (UTIAS) combustion lab for gaseous and liquid fuels at elevated pressures.

Fuels investigated for its sooting propensity include methane [7], biogas [8], ethylene [9] and n-

heptane [10]. In addition, the computational framework used in this study has been previously

applied to investigate the sooting propensity for several gaseous fuels and biogas [9,11–13]. Soot

formation and oxidation processes are dependant on several parameters. As a result, systematic

fundamental studies of simple small-scale non-premixed laminar flames are essential in order to

develop accurate models neccessary to study high-pressure turbulent flames.

Though experimental and numerical studies on the sooting propensity of methane, biogas and

other fuels [11] have been studied in the past, no such equivalent study has been done for

ethanol. This thesis therefore includes a numerical investigation of the sooting propensity of

ethanol as a function of pressure using laminar co-flow diffusion flames as the case study. The

numerical results are then compared qualitatively to the partial results obtained previously

by Karatas [14] within UTIAS combustion lab. However, due to high uncertainties of the

experimental setup to handle liquid fuels, no direct measurements of soot formation and flame

were available for quantitative comparisons to the numerical predictions.

1.2 Background

1.2.1 Ethanol

Ethanol is a very important energy carrier that can be produced from renewable energy sources.

It can be used as a fuel extender, octane enhancer and as an alternative fuel to replace reformu-


lated gasoline [15]. Recent developments suggest that ethanol can be derived more efficiently

from renewable sources thus reducing dependencies on fossil fuel energy sources [16].

There is a growing interest worldwide to find out new and cheap carbohydrate sources for

production of ethanol [17]. Currently, a heavy focus is on bio-fuels made from crops, such as

corn, sugar cane, and soybeans, for use as renewable energy sources. Though it may seem

beneficial to use renewable plant materials for bio-fuel, the use of crop residues and other

biomass for bio-fuels raises many concerns about major environmental problems, including

food shortages and serious destruction of vital soil resources [18]. For a given production line,

the comparison of the feedstocks includes several issues. This includes chemical composition of

the biomass, cultivation practices, vailability of land and land use practices, use of resources,

energy balance, emission of greenhouse gases, acidifying gases and ozone depletion gases. In

addition, absorption of minerals to water and soil, injection of pesticides, water requirements

and water availability are some other concerns which need to be addressed.

Ethanol feedstocks can be divided into three major groups. Which are sucrose-containing

feedstocks (e.g. sugar cane, sugar beet, sweet sorghum and fruits), starchy materials (e.g. corn,

milo, wheat, rice, potatoes, cassava, sweet potatoes and barley), and lignocellulosic biomass

(e.g. wood, straw, and grasses).

In the short-term, the production of ethanol as a vehicular fuel is almost entirely dependent on

starch and sugars from existing food crops [19]. The drawback in producing ethanol from sugar

or starch is that the feedstock tends to be expensive and demanded by other applications as

well [20]. Any ethanol project attacks seven major national issues: This includes sustainability,

global climate change, biodegradability, urban air pollution, carbon sequestration, national

security, and the farm economy. Lignocellulosic biomass is envisaged to provide a significant

portion of the raw materials for ethanol production in the medium and long-term due to its

low cost and high availability [18].

1.2.2 Soot Formation, Oxidation and Modelling

The complex nature of hydrocarbon-based reacting flows that involve many different physical

processes prevent a complete understanding of the soot formation and oxidation process. Cur-

rently, the least understood aspects of soot formation are the intermediate steps leading up to

the formation of the first soot particles [11]. However, the formation and destruction of soot

are relatively more well defined.

During combustion, a hydrocarbon fuel is pyrolyzed to form a smaller hyrocarbon, for example


acetylene, which react and form small aromatic compounds. Primary soot particles first appear

after the formation of soot precursors such as polyacetylenes and polycyclic aromatic hydro-

carbons (PAH). These particles react at the particle surface, increasing in size, colliding with

each other and forming new larger particles called agglomerates. During the formation of soot

particles, precursors and soot are also oxidized by oxygen-containing species. The oxidation

process continues until the soot particles are completely oxidized or diffused out of the flame

envelope [11]. The whole conversion process from gas to solid is largely chemistry controlled

and occurs over several miliseconds [21].

The interactions of different physical processes which occur over a wide range of spatial and

temporal scales pose significant challenges to the modelling of soot in flows. Detailed models of

soot formation includes three main components: gas phase kinetic mechanism, soot chemistry

sub-model and an aerosol dynamics model. The first model describes the formation of soot

precursors and large cyclic hydrocarbon molecules. The second model describes gas-particle

conversion which includes nucleation of primary soot particles and any reactions which occur

between the gas and particle surface. Finally the aerosol dynamics model, is used to describe

the inter-particle collision as well as any heat and mass transfer between the gas and the solid

phase [11]. Due to detailed nature of soot modelling, empirical soot models are adopted in

numerical simulations to remain computationally tractable. The semi-empirical model that has

been adopted in this study is an acetylene-based soot model [22]. Such models use a simple

kinetic mechanism based on acetylene as the sole precursor responsible for the nucleation and

growth of soot particles. Surface growth was assumed first order in acetylene concentration and

is dependant upon some function of the aerosol surface area. Two transports equations for soot

mass fraction and soot particle number density are solved in this semi-empirical model [23–25].

Soot formation and oxidation strongly affects the structure and stability of laminar diffusion

flames by enhancing radiation and altering local temperature fields. Reaction rates are highly

dependant upon temperature and therefore local gaseous species concentrations are strongly

influenced by the presence of soot. Thus, fully understanding the soot formation process in

laminar diffusion flames is essential for various engineering applications [12]. Previous experi-

mental and numerical studies on soot formation that were conducted for various fuels include

methane-air flames at pressures from 10 to 60 atm [26], syngas-air, syngas-methane and biogas-

air flames from 5 to 20 atm [27] and ethylene-air flames from 0.5 to 5 atm [12]. Some important

observations were made regarding the validity of the semi-empirical soot model adopted in the

numerical simulations. Overall, the predictions for soot volume fraction and temperature dif-

fered significantly from measurements. However, the soot model was still able to capture the

effects of pressure and predict the correct trends with reasonable accuracy. The discrepancies


were mainly attributed to the errors introduced by the acetylene-based soot model and uncer-

tainties in the boundary conditions. For instance, applying an adiabatic boundary condition

which matched the experimental results the closest, over-predicted the temperatures near the

tube wall and resulted in high soot concentrations low in the flame [11].

1.2.3 Effects of Pressure

Soot formation and oxidation processes are highly dependant on the flame environment. One

of the most important enviroment parameter is pressure. Pressure affects the flame structure

and other flame properties in both premixed and non-premixed flames. Though most practical

combustion devices operate at elevated pressures, our understanding of pressure’s influence on

the flame structure is still inadequate [28].

In laminar premixed flames, density of the mixture is proportional to the pressure while diffusiv-

ity is inversely dependant on pressure. Thus, an increase in pressure limits diffusive processes

while increases the density of the mixture. As a result, the flame speed of the fuel is re-

duced [29,30]. The adiabatic flame temperature however is rather independent of pressure and

is therefore unaffected, except near stoichiometric conditions where the temperature can go up

by approximately 50-100 K for a significant increase in pressure.

Conversely for laminar diffusion flames, pressure affects the shape and structure of the flame.

This is because pressure directly affects the chemical kinetics and buoyant forces [31]. Since

buoyancy induced acceleration is proportional to the pressure squared, an increase in pressure

will change the shape of the flame. In high pressure laminar diffusion flames, the streamlines

contract towards the center, due to buoyancy effects, thereby decreasing the diameter of the

flame [32]. It has also been suggested that the decrease in flame diameter at elevated pressures

could be due to increases in reaction rates, which in turn are caused by higher temperatures

and steeper concentration gradients [32,33].

Previous theoretical studies have shown that to a first-order approximation, flame height should

be independent of pressure and should only depend on mass flow rate [34, 35]. It was thought

that residence time, and thereby the axial velocity along the flame centreline was independent

of pressure, based on experimental measurements [33] and numerical predictions [36]. The

predictions were based on findings that the flame radius is proportional to p−0.5 [37,38], which

implies that centreline velocity and thereby residence time should be independent of pressure.

If the residence time is indeed not dependent on pressure, then flame height should be constant

at all pressures. While many previous experimental studies have indicated that this may not be


true, Joo [26] and Gulder [26] and Bento et al. [39] have clearly demonstrated that the visible

flame heights are indeed constant across a wide range of pressures. Numerically, Charest et

al. [6, 11] as well as Liu et al. [36] have also shown that flame heights are essentially constant

with pressure for fixed fuel mass flow rates.

1.3 Objectives

1.3.1 Numerical Investigation of One-Dimensional Laminar Premixed Flames

Before investigating the sooting propenstity of ethanol, it is important to first determine some

key fundamental combustion properties of the fuel. A preliminary investigation of ethanol

properties was carried out herein in which solutions of unstrained, planar, one-dimensional (1D),

laminar, premixed flames were computed and analyzed using an open source software package

known as Cantera [40]. Chemical kinetic mechansims of varying levels of detail designed for

modelling the combustion of ethanol fuel were considered, as described in Chapter 2. The

premixed flames solutions were determined and numerical solution of the laminar flame speeds

and adiabatic flame temperatures were obtained as a function of equivalence ratio and pressure.

The flame speed and temperature are key combustion properties that reveal much about nature

of the fuel [31].

Knowledge of flame speed gives an overall summary of some of the important characteristics of

the fuel such as reactivity, diffusivity and exothermicity and is critical in the design of practical

combustion devices [41]. In addition, flame speed also reveals the susceptibility of a flame fed

by a particular fuel mixture to flashback or extinguish [42].

The adiabatic flame temperature is the equilibrium temperature of the combustion products

assuming that no work is done at constant pressure [31]. It is an important parameter in quan-

tifying the heat release associated with the combustion of the fuel. The flame temperature can

also be a useful indicator of pollutant formation. In addition, the adiabatic flame temperature

also has an influence on flame propagation and extinction. Fuels with high flame temperatures

have higher heating values, which consequently raises the flame speed of the fuel [30].

Numerical simulations of the 1D premixed flames and the resulting flame speeds and tempera-

tures also provide an excellent way of examining the chemical kinetic mechanisms, both detailed

and reduced, and validating them against experimental data as well as other previous, more es-

tablished, kinetic mechanisms [43]. Converged solutions for the 1D premixed flame simulations

provided values for the following properties as a function of equivalence ratio and for a range


Figure 1.1: UITAS high-pressure laminar co-flow diffusion flame burner apparatus.

of pressures from 1 to 25 atm for ethanol using full Dryer [16] and Reduced Dryer [44] chemical

kinetic mechanism:

• flame speed;

• adiabatic flame temperature;

Cantera was used in evaluating all of these quantities and its usage is further detailed in Chapter

2 while the results of the numerical computations for the premixed flames described above are

presented and discussed in Chapter 3.

1.3.2 Numerical Simulation of Two-Dimensional Axisymmetric

Laminar Co-Flow Diffusion Flames

Though most combustion devices involve high pressure turbulent flames, there is no complete

framework of understanding flames in this regime due to experimental limitations, complex

flame geometries, complex physical and chemical interactions and large disparity in time and

length scales [6]. For these reasons, laminar flames are often studied. Since the turbulent flames

can be thought of as interactions of infinitesimal laminar flame structure, understanding the

interactions in laminar case is crucial to the understanding of the complex interactions for the

turbulent case [31].

As a part of this thesis, two-dimensional (2D) laminar coflow diffusion flames simulations of

the UTIAS high-pressure co-flow burner (shown in Figure 1.1) were be carried out using the


in house Computational Framework for Fluids and Combustion (CFFC) code. In particular,

the code will use the 2D flame solution method for primitive variables, Flame2Dp which was

developed by Charest et al. [11,45] which in turn was built upon the previous work of Northrup

and Groth [46]. This computational framework is a highly-scalable combustion modelling tool

which has been developed specifically for use on large multi-processor, high-performance and

parallel computer systems. The numerical framework uses a parallel-implicit time marching

scheme with a Newton-Krylov iterative solution procedure, a second order finite volume spatial-

discretization scheme and a block-based adaptive mesh refinement (AMR). The aim is to solve

the unmodified compressible form of the Navier-Stokes equations governing multi-species reac-

tive flows on 2D axisymmetric domains using multi-block, body fitted, quadrilateral meshes.

Piecewise linear limited reconstruction and Riemann-solvers’ flux evaluation, applicable to all

flow speeds, are used. An added challenge is faced due to large differences between spatial and

temporal scales which leads to numerical stiffness. The stiffness is also enhanced significantly at

very low Mach numbers when the disparities between acoustic and convective velocities become

large. When upwind schemes are applied in low Mach number cases, excessive artificial dissi-

pation is introduced, thus affecting the overall solution accuracy [47]. To overcome the stiffness

introduced by low speed flows, low-Mach-number preconditioning is adopted. In addition, the

framework features detailed treatments of chemical kinetics, radiation transport and soot for-

mation and transport. Radiation is modelled using the discrete ordinates method (DOM) [48]

to solve the radiative transfer equation and spectral absorption coefficients for gas band ab-

sorption are approximated using the wide-band model developed by Liu et al. [49] which is

based on the statistical narrow-band correlated-k (SNBCK) method [49, 50]. Furthermore, a

simplified semi-empirical model is employed to predict chemistry, nucleation, surface growth,

coagulation, and oxidation of soot particles [51, 52]. The latter assumes that acetylene is the

primary precursor leading to the formation of soot particles. Charest et al. [7,11–13], have ap-

plied this semi-empirical soot model to numerically investigate gaseous fuels such as methane,

ethylene and biogas.

In this thesis, numerical simulations of two-dimensional laminar co-flow diffusion flames were

performed for ethanol at pressures of 1, 5, 10 and 15 atm using two chemical kinetic mechanisms.

The full Dryer [16] and Reduced Dryer [44] chemical kinetic mechanisms were both considered.

The numerical solutions for the axisymmetric flames were compared against partial experimen-

tal results from UTIAS combustion lab to assess and quantify the differences between the model

and actual experiments. Though only experimental results were available for 1 atm and par-

tially for 5 atm, this numerical investigation is a useful first step and helpful in understanding

the general trends of diffusion flame structure and sooting propensity of ethanol. Solutions


provide predictions of the following properties for ethanol:

• flame structure (height, width etc.);

• temperature contours; and

• soot concentration;

The computational framework for laminar flames and its usage are discussed in further detailed

in Section 2.2, while the data and results of the laminar co-flow diffusion flame computations

are presented in Chapter 4.

Chapter 2

Numerical Solution Methods

2.1 One-Dimensional Laminar Premixed

Flame Modelling

As mentioned in Chapter 1 of this thesis, Cantera was used to evaluate the flame speeds and

adiabatic temperatures of ethanol liquid fuel as a function of pressure. Cantera is a suite

of object-oriented software tools for problems involving chemical kinetics, thermodynamics,

and/or transport processes [40]. It is a C++ based code with interfaces for python, Matlab, C,

and Fortran 90.

In the present study, two different existing mechanisms were considered when determining the

solutions of laminar premixed flames for ethanol. The mechanisms that were considered herein

are as follows:

• the Dryer mechanism [16];

- 238 elementary reactions

- 39 species

• the Reduced Dryer [44];

- 154 elementary reactions

- 29 species

- reduced from the Dryer mechanism

The Dryer mechanism was developed for combustion of ethanol and blends of ethanol with stan-

dard gasoline in gasoline engines. The mechanism includes low and intermediate temperature

chemistry under knock-prone conditions [16]. While the reduced Dryer mechanism is based on

10

Chapter 2. Numerical Solution Methods 11

the Dryer mechanism and developed for reduced computational cost [44].

2.1.1 Cantera Solution Method for Flame Speed

and Temperature

Cantera solves the steady state form of the conservation equations for a reactive gaseous mixture

in one-dimension (1D) to compute the laminar flame speed [29]. The conservation equations

of interest are continuity, species conservation and energy conservation equations. For steady

planar one-dimensional flows, the conservation equations can be expressed as:

dm′′

dx= 0, (2.1a)

m′′dYidx

+d

dx(ρYivi,diff ) = ωiMWi, for i = 1, 2, .., N species, (2.1b)

m′′cpdT

dx+

d

dx(−kdT

dx) +

N∑i=1

ρYivi,diffcp,idT

dx= −

N∑i=1

hiωMWi. (2.1c)

Here m′′i is the mass flux, Yi is the species mass fraction, vi,diff is the species diffusional velocity,

ωi is the species reaction rate and lastly MWi is the species molar mass. The flame speed Sl is

related to m′′ by using Sl = m′′/ρ.

In order to close the governing equations, Cantera employs the ideal-gas equation of state

and temperature dependent polynomials for species properties. In addition, a chemical kinetic

mechanism is required to provide the values for source term ωi. Feature of the numerical method

emplyed by Cantera is that the discretized form of the governing equations are solved using a

hybrid Newton/time-stepping algorithm to accelerate convergence [53].

Starting with a one-dimensional mesh, the mesh is authomatically refined locally in the region

with high gradients until an accurate steady-state premixed flame solution is achieved. In this

study, the two mechanisms for ethanol required 1000 grid points to obtain sufficiently accurate

converged solutions.

In the discretized form of the equations, the nth equation of the jth point in the 1D mesh is in

the form

F j,n(φ) = 0, (2.2)

where F j,n only depends on the solution at the points j-1, j, j+1. This produces a non-

linear coupled system of algebraic equations. The resulting residual equations are solved with

a modified Newton’s method. The initial step in solving the system of equations employs the


classical Newton’s method, which linearises around an initial solution estimate, φ0, and is given

by

F 0lin,n = Fi(φ

0) +∑j

∂Fi∂φj

∣∣∣∣φ=φ0

(φj − φ0j ), (2.3)

then the linear problem is solved to generate a new estimate for φ given by

F lin(φ1) = 0, (2.4a)

φ1 = φ0 − [J0]−1F 0, (2.4b)

where J is the system Jacobian, ∂Fi/∂φj .

If the Newton iterations fail to find a steady-state solution to the premixed flame of interest,

an attempt is made to solve a pseudo-transient problem, which potentially contains a larger

domain of convergence [53]. The pseudo-transient problem is created by adding transient terms

in each conservation equation, where physically reasonable, as shown below:

Adφ

dt= F (φ), (2.5a)

F (φn+1)−Aφn+1 − φn

∆t= 0. (2.5b)

The quantity A in Equation (2.5a) is a diagonal matrix with entries of 1 on the diagonal for

the equations with a transient term, and 0 for the constraint equations.

In summary, the algorithm of the C++ program in Cantera that solves for the one-dimensional

laminar flame speed and temperature proceeds as follows :

1. input the chemical mechanism, fuel-oxidizer mixture and environment condition data;

2. calculate unburned mixture properties;

3. determine equilibrium conditions and burned gas properties;

4. create the grid;

5. use provided initial guess of temperature and velocity profiles;

6. call inbuilt newton solver subroutine to iteratively improve upon the initial guess;

7. refine grid and call solver subroutine again if necessary; and

8. output to Tecplot (commercial plotting tool) for plotting of final solution data.


2.2 Axisymmetric Laminar Co-Flow Diffusion

Flame Modelling

The numerical solution procedure used in this study to simulate laminar reacting flows with de-

tailed chemistry, non-gray radiative heat transfer and soot is based on the numerical framework

developed by Charest et al. [6, 11]. This computational framework is a highly-scalable com-

bustion modelling tool which has been developed specifically for use on large multi-processor

high-performance parallel computer systems. The framework solves the conservation equation

for multi-species compressible reacting flows with soot on axisymmetric domains using multi-

block, body-fitted, quadrilateral meshes. Soot was modelled using a simplified semi-empirical

model approach proposed by Liu et al. [22]. The model describes the chemistry, nucleation,

surface growth, coagulation, and oxidation of soot particles [51, 52]. The latter assumes that

acetylene is the only precursor leading to the formation of soot particles.

The governing conservation equations are solved numerically utilizing a finite-volume scheme

developed by Groth and co-workers [54, 55] using the semi-discrete approach. Piecewise linear

limited reconstruction and Riemann-solver-based flux functions are used to evaluate inviscid

fluxes [56]. Viscous fluxes were determined using a second-order diamond path integral method

developed by Coirier and Powell [57]. Low-Mach-number preconditioning of both the inviscid

flux and the temporal derivative are applied in order to permit the efficient and accurate solution

of the conservation equations for low-speed flows associated with laminar flames [47]. The non-

linear, coupled ordinary differential equations (ODEs) were then solved to steady-state using

the block-based parallel implicit time-marching scheme in conjunction with a Newton-Krylov

iterative method [54].

In modelling radiation emitted and absorbed by soot and some participating gases, a discrete

ordinates method (DOM) coupled with point-implict finite volume approach was employed [58].

The spectral absorption coefficients for gas band absorption are approximated using the wide-

band model developed by Liu et al. [49] which is based on the statistical narrow-band correlated-

k method [49,50]. In addition, Cantera was employed to evaluate thermodynamics and transport

properties along with gas-phase kinetic rates.

The computational framework adopted in this study for laminar flames was previously applied

succesfully to study the effects of both high pressure and low gravity on flame structure and

sooting propensity for several gaseous fuels and biogas [9,11–13]. For the purposes of complete-

ness, the key aspects of the computational framework for laminar flames of Charest et al. [6,11]

are summarized below.


2.2.1 Governing Equations

As mentioned above, the computational framework used in this study solves the compressible,

2D axisymmetric form of the Navier-Stokes equations for a multi-component of thermally-

perfect reactive gaseous mixture. The governing equations in this case are as follows:

∂ρ

∂t+∇ · (ρv) = 0, (2.6a)

∂

∂t(ρv) +∇ · (ρvv + pI) = ∇ · τ + ρg, (2.6b)

∂

∂t(ρe) +∇ · [ρv(e+

p

ρ)] = ∇ · (v · τ )−∇ · q + ρg · v, (2.6c)

∂

∂t(ρYk) +∇ · [ρYk(v + V k)] = ωk, for k = 1, ..., N , (2.6d)

where t is the time, v is the velocity vector, ρ is the density of the mixture, p is the mixture

pressure, e is the mixture energy, τ is the fluid stress tensor, g is the gravitational acceleration

vector, q is the heat source flux vector, Yk is the mass fraction of the kth species, Vk is the

diffusional velocity of the kth species and ωk is the time rate of change of the kth species. The

ideal gas law is used to calculate the mixture density. The heat flux vector, q, is given by

q = −κ∇T + ρ

N∑k=1

hkYkV k − qrad, (2.7)

where κ is the thermal conductivity of the mixture, h and Y are the individual species enthalpy

and mass fractions respectively while qrad is the radiation heat flux. The gas phase diffusion

velocity vector for the kth species is given by

V k = −Dk

Yk∇Yk, (2.8)

where Dk is the species averaged diffusion coefficient.

2.2.2 Finite-Volume Solution Discretization Method

The coupled system represented by Equations (2.6a) - (2.6d) of the reactive gas mixture are

solved numerically using the parallel, implicit and finite-volume based scheme developed pre-

viously by Groth et al. [54, 55] and subsequently improved by Charest et al. [6, 11]. In this

approach, the domain of the physical problem is discretized into a number of finite-size compu-

tational cells and the integral form of the conservation equations is then applied to and solved

for the cells. For a general quadrilateral cell, (i, j), in a two-dimensional domain as shown in


Figure 2.1: Schematic diagram showing generic 2D quadrilateral computational cell.

Figure 2.1, the semi-discrete, non-linear and coupled form of the governing equations for the

primitive, cell-averaged solution quantities, W , are as follows:

dW i,j

dt=∂W

∂U

∣∣∣∣ij

·

− 1

Aij

∑face,k

(F k · nk∆lk)ij + Sij

(2.9)

where U ij and W ij are the cell averaged conserved and primitive representation of the solution

vectors. In Equation (2.9), Aij is the cell area, nk and ∆lk are the normal vector and edge

length for the kth face respectively, and Sij is the source term that includes contributions

from chemistry, gravity and boundary conditions. The flux, F k, includes both the inviscid and

viscous fluxes contributions (F and F v respectively).

2.2.3 Low-Mach-Number Preconditioning

Obtaining a solution of the non-linear system of ODEs of Equation (2.9) above resulting from

the finite-volume spatial discretization at low mach numbers has an added difficulty due to

the stiffness induced by the disparate velocity scales of the acoustic and convective waves [59].

Hence preconditioning is employed to reduce the numerical stiffness of the problem. Precondi-

tioning replaces the physical time derivatives with artificial derivatives in order to better match

the magnitudes of the acoustic and convective waves. The computational framework used

herein adopts the preconditioning scheme developed by Weiss and Smith [47]. The modified


conservation equations for an axisymmetric co-ordinate system can be expressed as:

Γ∂W

∂t+∂F

∂r+∂G

∂z=∂Fv

∂r+∂Gv

∂z+ S, (2.10)

where Γ is the preconditioning matrix as developed by Weiss and Smith. The eigenvalues of

the preconditioned Jacobian matrix in the r-direction, Γ−1 ∂F∂W , are as follows:

λ = [u′ − a′, u, u, u′ + a′, u, ..., u]T , (2.11)

for which

u′ = u(1− α), (2.12a)

a′ =√α2u2 + V 2

p , (2.12b)

α = 0.5(1− βV 2p ), (2.12c)

β = ρp +ρT (1− hp)

ρhT. (2.12d)

The subscripts in the expressions above represent partial derivatives of the quantities of interest.

For a perfect gas, ρT = −ρ/T , ρp = 1/(RT ), hp = 0, and hT = cp. The so-called preconditioned

sound speed, Vp, can be defined as

Vp = min[max(Vinv, Vpgr, Vvis,Mref · a), a], (2.13)

where a is the sound of speed and Mref is a reference Mach number used to prevent singularities

at stagnation points. A value of 10−4 for Mref was used in all the simulations conducted herein.

The remaining subscripted terms are the inviscid, pressure-gradient, and viscous velocity scales

as derived in [47] and given by

Vinv =√u2 + v2, (2.14a)

Vpgr =

√|∆p|ρ

, (2.14b)

Vvis =µ

ρ∆x, (2.14c)

where ∆p is the pressure gradient within the cell and ∆x is the length of the computational

cell.

2.2.4 Round-Off Error Control

Previous study by Charest et al. [6,11] determined that below Mach numbers of 10−3, machine

round-off errors began to contaminate the solutions for pressure. In order to fix this issue,

the solution method was modified to use the procedure developed by Choi and Merkle [60].


A reference pressure, p0, is used to reduce the influence of round-off errors in pressure at low

Mach numbers. Due to the modification, the new overall pressure, p, is then given by

p = p0 + p′, (2.15)

where p0 is the reference pressure and p′ is the deviation from the local pressure from p0. The

reference pressure is subtracted from Equation (2.6b) and p is replaced by p′ in the numerical

solution vector, W .

2.2.5 Inviscid Flux Evaluation

A second-order upwind Godunov scheme is employed to evaluate the inviscid numerical fluxes

at cell interfaces. Godunov’s method assumes that the solution within each computational

cell is piecewise constant and that the solution at the cell interface can be approximated via

upwinding [61]. Upwinding also preserves the monotonicity of the solution.

Given left and right solutions states, WL and WR, the flux at a cell interface is defined in

terms of a function that involves the solution of a Riemann problem, in a direction along the

cell interface normal, n. The numerical flux is given by

F · n = F(WL,WR, n), (2.16)

where the evaluation of F requires the solution of the Riemann problem. Roe’s approximate

Riemann solver [62] is used in the current solution method to solve the Riemann problem and

evaluate the fluxes. Harten’s correction [63] is also employed with the solver to rectify violations

of the entropy condition at sonic points. The flux in one direction at a cell interface is given by

F (R (WL,WR)) =1

2(FR + FL)− 1

2|A|∆W , (2.17)

where FL and FR are the left and right fluxes as functions of the left and right primitive solution

states WL and WR, ∆W = WR - WL. |A| = R|Λ|R−1 where R is a matrix composed of

primitive right eigenvectors and Λ is the eigenvalue matrix. The matrix A is the linearised flux

Jacobian evaluated at a reference state W . The reference state is typically chosen such that

Roe’s conditions are relaxed for reacting flows [64]. Therefore, the Roe-averaged flow variables

are given by

u =

√ρRuR + ρLuL

ρR + ρL, (2.18)

where uR and uL are either of the flow variables u, v, h, Yk. Additionally, ρ, the Roe averaged

density, is given by√ρRρL.


To control the dissipation at low Mach numbers that results with the use of upwind methods,

Equation (2.17) can be re- derived by using the pre-conditioned wave speeds as shown by Weiss

and Smith [47]. The linearised flux Jacobian term (with the |A| matrix) in Equation (2.17) is

modified as shown below.

|A|∆W ' A∆W = Γ

(Γ−1

F

W

)∆W = Γ|AΓ|∆W , (2.19)

where |AΓ| = RΓ|ΛΓ|RΓ−1. The subscript Γ shows that the matrix eigenvectors and eigenval-

ues are derived using the preconditioned system.

2.2.6 Higher-Order Spatial Accuracy for Inviscid Fluxes

Godunov’s method is first-order accurate and as a result the scheme can be excessively dissipa-

tive. To extend Godunov’s method to second or higher orders can be challenging since schemes

with constant coefficients will have non-monotonic behaviour near solution discontinuities [56].

In the computational framework of Charest et al. [6,11] the solution at the cell interface between

two adjacent cells is reconstructed by a piecewise linear function to achieve second-order spatial

accuracy. Slope limiters are employed to ensure that monotonic (non-oscillatory) behaviour of

the solution is preserved at or near discontinuities.

The slope limiters function by locally reducing the reconstructed solution to first-order, thereby

damping out any over- or under-shooting behaviour at discontinuities. The reconstructed left

and right solution values at a cell interface given by the piecewise linear limited representation

for an interface (i + 12 , j) are given by

WL = W ij + φij

[∂W

∂r

∣∣∣∣ij

(ri+ 12,j − rij) +

∂W

∂z

∣∣∣∣ij

(zi+ 12,j − zij)

], (2.20)

WR = W i+1,j + φi+1,j

[∂W

∂r

∣∣∣∣i+1,j

(ri+ 12,j − ri+1,j) +

∂W

∂z

∣∣∣∣i+1,j

(zi+ 12,j − zi+1,j)

], (2.21)

where φ is the slope limiter. Slope limiting is performed using methods designed especially

for multiple dimensions [56]. The cell gradients are calculated using linear reconstruction from

Green-Gauss theory as discussed by Barth and Jespersen [65].

2.2.7 Viscous Flux Evaluation

The computational framework in this study employs the centrally-weighted diamond-path

method, as developed by Coirier and Powell [57], to evaluate the viscous fluxes at the boundaries


Figure 2.2: Schematic diagram showing diamond path viscous flux reconstruction stencil for a

quadrilateral computational cell.

of each computational cell. The viscous flux for a cell face is given by

Fv · n = G(W ,∇W ,n), (2.22)

where G is the viscous flux function.

Figure 2.2 shows the diamond-path method. Gradients at each face are calculated by applying

the divergence theorem along the path. The solution is known at the cell-centred vertices,

however the solution state at the vertices (nodes) of the cell must be interpolated. A weighting

scheme developed by Zingg and Yarrow [66] that linearly reconstructs nodal data using cell-

centred solution data of the neighbouring cells is used.

2.2.8 Steady State Relaxation Method

The computational framework for laminar flames uses Newton’s method to numerically solve

the non-linear algebraic equations resulting from the spatial discretization of the steady form of

the governing conservation equations given above. In particular, the semi-discrete form of the

governing equations are iteratively relaxed to converge to a steady-state solution, W , satisfying

R(W ) =dW

dt= 0. (2.23)


Charest et al. [6,11] developed the Newton algorithm which is employed in this computational

framework for laminar flames. Their approach follows the scheme originally developed by Groth

and Northrup [67]. The implementation of the algorithm makes use of a Jacobian matrix free,

inexact Newton method along with an iterative Krylov subspace linear solver.

Starting from an initial estimate for the solution, W 0, successively improved estimates of the

solution to the equation above can be found by solving the linear system for ∆W n.(∂R

∂W

)n∆W n = J(W n)∆W n = −R(W n), (2.24)

where J is the residual Jacobian and n is the step number. The improved solution at the next

step, n+ 1, is found by using

W n+1 = W n + ∆W n. (2.25)

This iterative process is repeated until a suitable reduction in the residual norm is achieved

and the condition ||R(W n)|| = ε||R(W 0)|| is satisfied; ε is a tolerance that is set to 10−7 in

the solution method.

Newton’s method, at each step, requires the solution of the linear system Jx = b where

x = ∆W and b = −R(W ). For a system of non-linear, coupled equations, the Jx = b

system is large, sparse and asymmetric. Such a system can be solved by using the generalized

minimal residual (GMRES) technique as developed by Saad and Schultz [68, 69]. GMRES is

an Arnoldi-based solution method which generates orthogonal bases of the Krylov subspace

to construct the solution. GMRES only uses matrix vector products at each step to create

the new trial Krylov vectors thereby greatly reducing the storage requirements for forming

J [70]. The GMRES iterations can be terminated by solving the system to some previously

specified tolerance, ||Rn + Jn∆W n)|| < χ||R(W n)||, where χ is set to 0.1 in the solution

method. Finally, to further lower memory requirements, the computational framework uses

a modified version of GMRES, GMRES(m), that restarts and refreshes the search directions

every m iterations.

2.2.9 Radiation Transfer Equation (RTE)

The radiation transfer equation (RTE) is the conservation law for the spectral intensity applied

to a monochromatic beam of light [71]. It is derived by applying an energy balance to a beam

of photons which are confined to a infinitesimal solid angle element and passing through an

infinitesimal volume of participating media [11]. In cylindrical coordinates, the steady form of

the RTE is a function of two dimensions(r,z) and two angular coordinates(θ,ψ) is


µ

r

∂

∂r(rIv)−

1

r

∂

∂ψ(ηIv) + ξ

∂

∂z(Iv) = −βvIv + kvIbv +

σsv4π

∫4πIv(s)Φv(s, s)dΩ, (2.26)

where Iv is the spectral intensity, Ibv is the blackbody radiative intensity, s is a unite vector,

wn is the wavenumber, kv is the absorption coefficient, σv is the scattering coefficient, and Φwn

is the scattering phase function. Also, β = kv + σsv is the extinction coeffient, θ and ψ are the

polar and azimuthal angles and µ, η and ξ are the direction cosines. These values define the

the unit vector s in the Cartesian coordinates which is given by:

s = µi+ ηj + ξk. (2.27)

In the absence of scattering, Equation (2.26) reduces to first-prder linear-hyperbolic differential

equations.

The next important quantity in solving for the heat flux term in Equation (2.6c) is the spectral

radiation flux vector, qv. It is defined as the net flow of radiant energy due to radiation from

all directions per unit area, time and wavenumber interval [11]. This is given by

qv =

∫ 4π

0sIv dΩ. (2.28)

Next, rearranging the RTE equation and integrating over Ω (all the solid angles), the divergence

of the heat flux vector can be expressed as

∇ · qv = kv[4πIbv −∫ 4π

0Iv dΩ] = kv[4πIbv −Gv], (2.29)

where Gv is the total incident radiation and is given by

Gv =

∫ 4π

0Iv dΩ. (2.30)

Since the radiative properties vary with v the wavenumber, Equation (2.29) must be integrated

over the entire spectrum. Thus the divergence of the the total radiation heat flux, ∇ · qrad in

the source term of Equation (2.6c) can be determined.

In this study the solution of the RTE is done using the DOM method which provides an excellent

balance between computational efficiency and accuracy. However, evaluating ∇ · qrad using the

DOM is costly due to the large number of unknowns associated with non-gray radiation. As

a result, solution of the RTE is decoupled from the gas-particle flow equations and solved

sequentially in a loosely-coupled fashion at each iteration. Refer to the study of Charest for

further details on the radiation modeling [11].


2.2.10 Discrete Ordinates Method (DOM)

Equation (2.26) is solved using the space-marching finite-volume approach outlined by [58].

The approach yields

µml (AEIE,ml −AW IW,ml) + ξml (ANIN,ml −ASIS,ml)

− (AE −AW )(αm,l+1/2IP,m,l+1/2 − αm,l−1/2IP,m,l−1/2

)/wml

= ∆V κP (IbP − IP,ml) ,

where the subscript P denotes quantities at the cell center, and the subscripts E, W , N and S

refer to quantities evaluated at the respective cell faces. For the sake of clarity, the quadrature

and band indices have been removed.

The cell volume, ∆V , and face areas, A, are as follows:

AN = AS = π(r2E − r2W ), (2.31a)

AE = 2π∆zrE , (2.31b)

AW = 2π∆zrW , (2.31c)

∆V = π(r2E − r2W )∆z, (2.31d)

where ∆r and ∆z are the cell-sizes in the r- and z-directions, respectively. For the purpose of

reducing the number of unknowns, the cell-edge intensities are related to the volume-averaged

intensity by the following expression:

IP,ml = γsIN,ml + (1− γs)IS,ml,= γsIE,ml + (1− γs)IW,ml, (2.32a)

IP,ml = γaIP,m,l+1/2 + (1− γa)IP,m,l−1/2, (2.32b)

where γs and γa are the spatial and angular differencing parameters respectively. For both

parameters, a value of 1 corresponds to upwind differences and 0.5 corresponds to central

differences. Central differences were used for both the spatial and angular discretization in all

of the computations.

Substituting into Equations (2.32a) and (2.32b) rearranging for the nodal intensity gives∆V κP + µmlAE/γs + ξmlAN/γs −

(AE −AW )αm,l+1/2

wmlγa

IP,ml =

∆V κP Ibp + µmlAEWIW,ml/γs + ξmlANSIS,ml/γs

− (AE −AW )[αm,l−1/2 + (1− γa)αm,l+1/2/γa

]IP,m,l−1/2/wml, (2.33)


where

AEW = (1− γs)AE + γsAW , (2.34a)

ANS = (1− γs)AN + γsAS . (2.34b)

Numerical solution of Equation (2.31a) proceeds as follows. First, the surface intensities and

internal source terms are estimated everywhere in the domain. The lower left corner of the

domain is chosen as a starting point so that all outgoing directions lie in the first quadrant

(i.e. µml > 0 and ξml > 0). Since the west and south faces of the control volume in this corner

are part of the enclosure surface, their intensities are specified by the boundary conditions.

From these known face values, IP,ml is computed using and the downstream intensities IE,ml

and IN,ml are determined from Equation (2.32a). One by one, the first-quadrant intensities

are calculated for all volumes in the enclosure. This procedure is repeated three more times

starting from the remaining corners of the enclosure and covering the other three quadrants

of directions. After sweeping all directions for IP,ml, the boundary values and radiative source

terms are updated. This procedure is repeated until convergence is met [11].

Solutions were deemed converged when the maximum change in the cell-averaged intensity

everywhere in the domain from one iteration to the next was less than a specified tolerance.

Throughout this thesis, the following convergence criterion was used:

∣∣In+1P − InP

∣∣ ≤ 10−12 for all P , m, l (2.35)

where the superscripts n and n+ 1 denote the iteration number.

2.2.11 Parallel Block-Based Solution Scheme for RTE

The DOM was solved in a parallel fashion at each time-step on the multi-block mesh along with

Equations (2.6a–2.6d) and Equations (2.36a–2.36b) by simultaneously sweeping all directions

on the domain local to each processor. Solution content was shared among the processors by

exchanging the state at the face-center of cells aligned with the block boundaries. Changes

in mesh resolution were handled by linearly interpolating the coarse-mesh solution onto the

fine-mesh and averaging the fine-mesh solution onto the coarse-mesh. Since the radiation solver

employs a space-marching technique, repeated numerical iterations are required to propagate

information from upstream boundaries to downstream blocks. As a result, a penalty in terms of

parallel efficiency was incurred because the number of iterations required to solve the radiation

field increased with the number of blocks.


2.2.12 Overall Solution Algorithm for RTE

The overall algorithm is summarized as follows:

1. Set initial conditions for W and I everywhere in domain.

2. Compute the spectral absorption coefficient for mixture.

3. Solve the RTE using the DOM described in Section 2.2.10 .

4. Update ∇ · qrad.

5. Solve Equation (2.23) for the gas/soot mixture, performing n Newton iterations.

6. Update primitive solution state W .

7. If not converged, return to step 2. The convergence criteria is defined in Section 2.2.8.

Throughout this study only one Newton iteration (n = 1) for was performed before updating

the radiation intensity field. Larger values of n up to five were tested but found to deteriorate

the performance of the overall solver. As n is increased, the CPU time required to advance the

solution a fixed interval ∆t in the computational domain decreases. However, increasing n also

increases the number of iterations required to obtain a converged, coupled solution.

2.2.13 Soot Modelling

Mathematical models describing the soot particles and their interactions with the gaseos mixture

are complex. In order to remain computationaly tractable, the mathematical representation is

simplified. This study simplies the aerosol representation by adopting a model based on the

solution of just two transport equations [23–25]. The transport equations are solved in terms

of number of soot particles per unit mass, NS and soot mass fraction Ys are

∂

∂t(ρYs) +∇ · [ρYs(v + VY )] = SY , (2.36a)

∂

∂t(ρNs) +∇ · [ρNs(v + VN )] = SN , (2.36b)

where v is the mixture velocity vector, VY and VN are soot diffusion velocities which include

contributions from both thermophoresis and Brownian motion. The variables SY and SN are the

source terms for Ys and Ns respectively due to nucleation, growth, oxidation and coagulation.

These source terms are defined in the following section.


The integration of these simplified transport equations will enable computation of the total

number concentration and volume fraction of the soot particles [49]. The simplified aerosol

description makes several assumptions. It assumes that the soot particles are perfectly spher-

ical and upon collision coalesce to form a new spherical particle that has the same equivalent

mass [49]. In addition, the spherical soot particles are also assumed to have constant composi-

tion and density.

However this approach has several shortcomings. The main shortcoming is the assumption that

the particles coalescing instantaneously upon collision. This has an affect on the soot particle

growth and oxidation rates since these quantities are highly dependant on particle surface area.

However more sophisticated treatment of the soot aerosol dynamics is beyond the scope of this

thesis.

2.2.14 Soot Chemistry

As mentioned before, the simplied soot kinetics as described by Liu et al. [49] model the

soot formation and oxidation through four steps - nucleation, surface growth, coagulation and

oxidation. Additionally acetylene is assumed to be the sole precursor for the presence of soot.

The resulting reactions describing these steps for soot are as follows:

C2H2 −−→ 2 C(s) + H2,

C2H2 + n · C(s) −−→ (n + 2 ) · C(s) + H2,

C(s) + 12 O2 −−→ CO,

C(s) + OH −−→ CO + H,

C(s) + O −−→ CO,

n · C(s) −−→ Cn(s).

From the chemical kinetic mechanism above, the source term in Equation (2.6c) can be ex-

pressed as

SY = 2MS(R1 +R2)− (R3 +R4 +R5)As, (2.38)

where MS is the molar mass of soot assumed to be equal to molar mass of carbon and R3, R4

and R5 are the soot oxidation reaction rates for O2, OH and O respectively. The terms R1 and

R2 are the soot nucleation and surface growth rates which is expressed as


R1 = k1[C2H2], (2.39a)

R2 = k2A0.5S [C2H2], (2.39b)

where rate constants k1 and k2 are [49]

k1 = 1000exp(−16103/T ), (2.40a)

k2 = 1750exp(−10064/T ). (2.40b)

While As is the soot surfaqce area per unit volume which describes the dependence of soot

surface growth. This quantity is related to soot mass and number density by this following

relation:

As = π(6

π

1

ρs

YsNs

)2/3(ρNs), (2.41)

where here ρs is the density of soot particles and is taken to be 1900kg/m3. The oxidation

rates R3, R4 and R5 are modelled by using [49]

R3 = 120(kap

1 + kz), (2.42a)

R4 = φOHk4T−12 pOH , (2.42b)

R5 = φOk5T−12 pO. (2.42c)

The source term in Equation (2.39a) describes the production and oxidation of soot particle

number density which includes nucleation and agglomeration. This source term is expressed as:

SN =2

CminNAR1 − 2Ca(

6Ms

πρs)1/6(

6kBT

ρs)[C(s)]1/6(ρNs)

11/6, (2.43)

where NA is Avogadro’s number, Cmin is the number of carbon atoms in the initial soot particle,

CA is the rate constant for agglomeration and C(s) is the molar concentration of soot. In this

study, coalescense was neglected by setting CA to zero and Cmin is set to 700 based on the

recommendations of Liu et al. [49].

Chapter 3

Results I: One-Dimensional Laminar

Premixed Flames

3.1 Benchmark Case

Before proceeding to the numerical study of some of the fundamental important combustion

properties of ethanol as a function of pressure, the validity of the chemical kinetic mechanisms

for ethanol were first explored by comparing the predicted laminar flame speeds to the existing

experimental data of Mansour et al. [72]. The experimental measurements for were conducted

flame speed data was done for pressures ranging from 1 atm and 5 atm for an initial mixture

temperature of 380 K at equivalence ratios from 0.7 to 1.4. The numerical predictions of laminar

flame speed were obtained using the Dryer [16] mechanism and Cantera as discussed in Chapter

2.

The comparison of the predicted laminar flame speeds from Cantera to the experimental mea-

surements of Mansour et al. [72] is depicted in Figure 3.1 for the Dryer mechanism. In general,

the predictions given in Figure 3.1 for Dryer mechanism are in rather good agreement with

experimental measurements. The mechanism is able to capture the correct trends for flame

speed as a function of equivalence ratio. The difference between the measured and calculated

value for laminar flame speed is within 10%. Accounting for experimental error, the calculated

values are in good agreement with the experimental values. In general for fuel lean conditions

conditions, the discrepancy is the greatest at around 10%. While for fuel rich conditions, the

experimental and calculated values closely match. The preceeding results are an indicator of

the level of accuracy of the mechanisms for ethanol considered here.

27

Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 28

Figure 3.1: Comparison of experimental data of laminar flame speed to predicted values from

Cantera for ethanol using the Dryer mechanism at 1 atm and 5 atm.

3.2 Laminar Flame Speed

3.2.1 Effects of Chemical Kinetic Mechanism

As mentioned in Chapter 2, two mechanisms for ethanol are considered in this work. Which are

the Dryer mechanism [16] and Reduced Dryer mechanism [44]. The reduced Dryer mechanism is

obtained by using the Directed Relation Graph (DRG) method proposed by Law et al [73]. This

systematic approach for mechanism reduction generates a skeletal mechanism by applying the

theory of DRG to identify unimportant species which in turn removes the reactions involving

the unimportant species. The theory quantifies the direct influence of coupling among the

species [73]. The reduced mechanism used here is generated using ignition delay calculations

as the benchmark [44].


Comparison of the predicted flame speed calculations from Cantera for ethanol using two dif-

ferent mechanisms are shown in Figures 3.2, 3.3, 3.4 and 3.5. The figures show the predicted

flame speeds for equivalence ratios between 0.5 – 1.8 or 2.0 and at all pressures from 1 to 25

atm using the Dryer [16] and Reduced Dryer mechanism [44]. Note that all of the premixed

flame simulations were conducted with an initial unburned mixture temperature of 300 K.

Overall, both of the mechanisms for ethanol seem do a reasonable job of predicting the laminar

flame speed, and provide generally similar results. The results differ to within 15% of each other

at the intermediate pressures and differ about 5% at the extreme pressures at 1 atm and 25

atm. Another notable feature of the reduced mechanism is that it consistently over-predicts the

flame temperature as compared to the Dryer mechanism. The difference is especially greatest

at fuel lean conditions.


Figure 3.2: Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values

from Cantera at 1 atm and 5 atm.


3.2.2 Effects of Stoichiometry and Pressure

Based on the results for the benchmark validation study discussed above and the comparisons

of the predicted flame speeds for ethanol using the two different mechanisms, it is evident

that both mechanisms yield rather similar results. Next the effects of fuel/air stoichiometry

and pressure on the predicted laminar flame speeds are evaluated for both mechanisms. The

predicted laminar flame speeds for equivalence ratios ranging from 0.5 to 1.8 and at pressures

of 1, 5, 10, 15, 20 and 25 atm are all shown in Figure 3.6.

It is evident from the Figure 3.6 the influence of stoichiometry on the flame speeds of ethanol.

In general, the flame speed increases with equivalence ratio from lean and reaches peak values

under slightly fuel rich conditions close φ = 1.1. As the mixture becomes fuel rich, flame speed

decreases.

The effects of pressure and equivalence ratio on the thermal diffusivities are evident for the

range of fuel compositions of interest. In theory, increased pressure results in a significant

reduction in the thermal diffusivity. This would account for with the observed reduction in

flame speed with an increase in pressure as seen in Figure 3.6.

Theoretical considerations suggest that the laminar flame speed, sL, is proportional to p−0.5 for

conventional hydrocarbon fuels [29]. To assess the validity of this result for the fuels in question,

the laminar flame speed data has been re-plotted as a function of pressure using log-log scales

for the plots. The results are shown in Figure 3.7. A line representing the expected theoretical

behaviour that sL ∝ p−0.5 is also shown in each plot for reference purposes.

As evident from the plots of the flame speeds for ethanol as a function of pressure depicted

in Figure 3.7, the expectation that the flame speed is indeed proportional to p−0.5 would also

seem valid here. The numerical curves for each equivalence ratio in these cases are in relatively

good agreement with the expected theoretical result. The agreement is best represented for

pressures between 1–10 atm and for leaner flames. At high pressures and equivalence ratios,

the exponent seems to become slightly higher for all of the fuels, which has been observed in

past experimental research by Okajima et al. [74].


Figure 3.6: Comparison of calculated values of laminar flame speed from Cantera for pressures

between 1 atm and 25 atm for Dryer and Reduced Dryer mechanism.


Figure 3.7: Laminar flame speed as a function of pressure comparison with theory.


3.3 Adiabatic Flame Temperature

Finally, the effect of stoichiometry and pressure on the adiabatic flame temperature is shown

in Figure 3.8 for the two mechanisms of ethanol. In general, the maximum flame temperature

occurs for slightly fuel rich conditions for all compositions with values ranging from 2200 K

to 2300 K at 1 atmosphere. This is explained by taking into account the balance of energy

required to increase the temperature of the combustion products (depending on the molar

specific heat of the products) and the energy lost during dissociation of the products at high

temperatures [30]. If dissociation were ignored, the peak of the flame temperature curve would

be at the stoichiometric point. Since dissociation is inhibited at high pressures, the temperature

peak at 25 atmospheres pressure corresponds with the stoichiometric point, as shown in Figure

3.8. In general, the flame temperature increase with pressure is minimal even at 25 atmospheres;

the increase in temperature was noted to be less than 100 K.


Figure 3.8: Adiabatic flame temperature of Dryer and Reduced Dryer mechanism as a function

of pressure.

Chapter 4

Results II: Laminar Co-Flow

Diffusion Flames for Ethanol

4.1 Introduction

The objective of the simulations described in this chapter was to investigate via a numerical

approach the sooting propensity of fully vaporized ethanol. This was accomplished by per-

forming simulations of laminar co-flow diffusion flames for this liquid biofuel for a range of

pressure conditions. The numerical simulations were carried out using a state-of-the-art lami-

nar combustion modelling framework developed by Charest et al. [6,11]. Numerical results were

obtained for both the reduced Dryer [44] and Dryer [16] chemical kinetic mechanisms. However,

due to current limitations of the experimental setup to vaporize ethanol, stable experimental

flames were only possible in the previous research by Karatas [14] for 1 atm and flickering

flames were obtained for 5 atm. For this reason, no experimental measurements of soot fraction

or temperature were available for comparison to the simulated results for the cases of interest

here. Only qualitative comparisons of a visual nature were possible between the experimental

and numerical results. This work simulates ethanol which is diluted with nitrogen by 70% by

volume. The fuel was also pre-heated to 400 K to mimic the experimental setup examined by

Karatas [14].

39

Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 40

Figure 4.1: University of Toronto Institute for Aerospace Studies (UTIAS) high-pressure lami-

nar co-flow diffusion burner schematic.

4.2 Experimental Setup

The experimental setup employs a laboratory-scale, high-pressure axisymmetric co-flow burner

developed at UTIAS [26, 38, 39]. The UTIAS burner consists of a central fuel tube with 3 mm

inner diameter and a concentric tube of 25.4 mm inner diameter that supplies the air as seen in

Figure 4.1. The outer surface of the stainless steel fuel nozzle is tapered to reduce the formation

of wakes behind the tube walls to improve flame stability [14]. Sintered metal foam is inserted

in the fuel and air nozzles to straighten the flow and provide as close as possible a uniform

velocity profile at the nozzle exit.

4.3 Discretization and Boundary Conditions

The computational domain for the co-flow diffusion flames in this work is shown in Figure 4.2.

The fuel jet is emitted from the center of the annular ring while the air mixture is emitted

from the outer ring. The computational domain extends radially outwards 20 mm and 25 mm

downstream of the fuel nozzle exit. The far-field boundary was treated using a free-slip condition

which neglects any shear imparted to the co-flow air by the burner walls. The modelled domain

is also extended 9 mm upstream into the fuel and air tubes to account for the effects of fuel

preheating observed by Guo et al. [75] and to allow uniform velocity distribution to be specified

at the inlet to better represent the inflow velocity distribution at the nozzle exit. Thus, uniform


Figure 4.2: Schematic of computational domain.

velocity and temperature profiles were specified for both the fuel and air inlet boundaries. At

the outlet, temperature, velocity and species mass fractions are extrapolated while the pressure

is held fixed.

A simplified representation of the fuel tube geometry was used in simulating the flame. The

tapered edge of the fuel tube in the experimental setup was approximated by a tube with 0.4

mm thick walls. The three surfaces that lie along the tube wall were modelled as adiabatic

walls with zero-slip conditions in this work. The computational domain was divided into 192

cells and 8 blocks in the radial and 320 cells and 64 blocks in the axial direction. The resulting

computational mesh (as shown in Figure 4.3) which contains 61,080 structured cells and non-

uniformly stretched. These computational cells were stretched towards the burner exit to

capture interactions close to the fuel tube walls and towards the centreline to capture the

high temperature gradients in the core flow of the flame. Charest et al. [6, 11] have previously

demonstrated that this level of mesh resolution is more than sufficient to accurately represent

the laminar co-flow diffusion flames associated with the UTIAS high-pressure burner of interest

here.


Figure 4.3: 2D computational grid showing the clustering of computational cells. The mesh

contains 61,080 cells.

4.4 Solution Procedure for Simulations

To obtain the numerical solutions for ethanol laminar co-flow flames using the computational

framework developed by Charest et al. [6, 11], the following steps were followed:

Step 1: The computational domain was initialized with cold air, fuel at 400 K, and a small

rectangular region above the fuel exit plane is used as an igniter to initiate the chemical reac-

tions. The igniter region is initialized with an unburnt fuel-air mixture temperature of 1800

K.

Step 2: Ignition can be a rather chaotic that includes a complex transient phase. A semi-

implicit relaxation scheme was used within the computational framework [46] to partially solve

Equations (2.6a) – (2.6d) and find an improved initial starting estimate for Newton’s method.

Due to the rather long transient chaotic transient phase for the ethanol flames, longer period

of semi-implicit iterations (approximately 600 000 iterations) was required in order to obtain a

suitable initial estimate for the implicit solver.


Thousands of Iterations

Den

sity

L2-

Nor

m

686 688 690 692 69410-4

10-3

10-2

10-1

100

101

102

103

104

105

Step 2

Step 3

Step 4

Step 5

Figure 4.4: Typical convergence history for ethanol-air flames. The normalized L2–norm of the

continuity equation is shown.

Step 3: The solution from step 2 was used as an initial guess to solve Equations (2.6a) – (2.6d)

using the fully implicit scheme as described in Section 2.2. As the solution converges to the final

solution, the value for Mref was decreased to 10−4 while the CFL value was slowly increased

to between 1–5.

Step 4: Using the fully converged solution from step 3 as the initial guess and the DOM

radiation model is employed without soot, the Equations (2.6a) – (2.6d) are solved once again.

Step 5: Obtain the final solution with soot. This results in another chaotic non-linear phase.

This phase was nursed by reducing the CFL to between 0.1–0.5 and then slowly increasing it

back up to 1–3.

Convergence was said to be achieved when the mass, momentum and energy residual L2–

norms were reduced by at least five orders of magnitude. Figure 4.4 shows sample of a typical

convergence history for the ethanol-air laminar co-flow flames, illustrating each of the simulation

phases outlined above.


Figure 4.5: Experiemental photographs of co-flow diffusion flames of approximately 5 mm in

height for ethanol at pressures of 1 atm and 5 atm as obtained by Karatas [14].

4.5 Experimental Results

Karatas [14] attempted to conduct the experiments for ethanol flames of interest; however, was

unsuccesful due to issues with the fuel vaporizer used in the experiments. Karatas was however

partially successful in that a stable flame was obtained at 1 atm and a slightly flickering flame

was observed at 5 atm. Visually, the flame did not soot at 1 atm while the flame produced

soot at a pressure of 5 atm. The experimental photographs of these two flames are shown

in Figure 4.5, with estimated flame heights of approximately 5 mm based on the luminous

regions of the flame. Due to the high uncertainties in the experimental setup for ethanol,

no measurements of soot volume fraction nor temperature were conducted. The experimental

values of 0.5 mg/s for fuel mass flow rate and 0.11 g/s for air mass flow rate were matched in the

numerical simulations to allow a qualitative comparison of the simulations to the experimental

images of the flames.


4.6 Predicted Temperature Field

4.6.1 Contours of Temperature

The predicted temperature contours for ethanol using two different mechanisms are shown in

Figure 4.6. In general both mechanisms predict approximately the same maximum temperature

and similiar temperature contours especially at higher pressures. In general the differences

between the two mechanisms for maximum temperature is less than 1.5%. Also, the predicted

maximum temperature of Reduced Dryer is consistently lower than Dryer mechanism. Referring

to Figure 4.6 again, the flame structure varies with increase in pressure. Flames at 1 atm exhibit

greater bulging while flames at higher pressures are narrower and more conical.

4.6.2 Radial Temperature Profiles

Three axial locations were chosen to compute the radial temperature profiles as seen in Fig-

ure 4.7 for the Dryer mechanism. The three locations chosen are: low in the flame where soot

undergoes nucleation, middle of the flame where it is close to the maximum soot volume fraction

and higher in the flame where soot is oxidized. In general, the peak temperatures moves closer

toward the centreline across all pressures.


Radius (mm)

Hei

gh

t (m

m)

-10 -5 0 5 10

-5

0

5

10

15

20

25T(K)

1900180017001600150014001300120011001000900800700600500400

DryerReduced Dryer

1 atm

1862.35 K1833.22 K

Radius (mm)

Hei

gh

t (m

m)

-10 -5 0 5

-5

0

5

10

15

20

25T(K)

1900180017001600150014001300120011001000900800700600500400

DryerReduced Dryer

5 atm

1973.01 K1972.07 K

Radius (mm)

Hei

gh

t (m

m)

-10 -5 0 5

-5

0

5

10

15

20

25T(K)

1900180017001600150014001300120011001000900800700600500400

DryerReduced Dryer

10 atm

2011.61 K 2013.87 K

Radius (mm)

Hei

gh

t (m

m)

-5 0 5

-5

0

5

10

15

20

25T(K)

1900180017001600150014001300120011001000900800700600500400

DryerReduced Dryer

2041.91 K

15 atm

2038.17 K

Figure 4.6: Effect of reaction mechanism and pressure on flame structure for ethanol at pressures

of 1, 5, 10 and 15 atm. Peak temperature is indicated on the bottom right and left corners.


Radius (mm)

Tem

per

atu

re (

K)

0 1 2 3 4

500

1000

1500

2000

0.5 mm1.0 mm1.5 mm

5 atm

Radius (mm)

Tem

per

atu

re (

K)

0 0.5 1 1.5 2 2.5 3 3.5 4

500

1000

1500

2000

0.5 mm1.0 mm1.5 mm

10 atm

Radius (mm)

Tem

per

atu

re (

K)

0 0.5 1 1.5 2 2.5 3 3.5 4

500

1000

1500

2000

0.5 mm1.0 mm1.5 mm

15 atm

Figure 4.7: Radial temperature profiles at three axial locations for ethanol for pressures 5, 10

and 15 atms using Dryer mechanism.


4.7 Prediction of Flame Heights

In this work, the stoichiometric mixture fraction was used to estimate the flame height of the

flames from numerical solutions. Given a steady state solution, the mixture fraction of the fuel

can be calculated for the entire domain by using the following equation [76]:

Z =

YC−YC,2

mWC+

YH−YH,2

nWH+

YO−YO,2

αWO

YC,1−YC,2

mWC+

YH,1−YH,2

nWH+

YO,1−YO,2

αWO

, (4.1)

where Y and W are the mass fraction and atomic weights of the carbon, hydrogen and oxygen

atoms, m and n are the number of carbon and hydrogen atoms in the fuel, 1 and 2 signify the

fuel and air inlets, and α is the stoichiometric fuel-air ratio.

From the given equation, the predicted flame height has been imposed on the temperature

contour plots in Figure 4.6. The peak temperature of the contours are indicated by the values

on the lower left and right corners as seen in Figure 4.6. In general, the flame height remains

constant across all pressures for both mechanisms at about 3.15 mm. As previously mentioned,

experimentally, the flame height at 1 and 5 atm was found to be approximately 5 mm, which

is very close to the predicted flame heights obtained in the present numerical simulations.

4.8 Prediction of Flame Radius

Previous experimental and numerical research with sooting diffusion flames has shown that the

flame radius proportional to p−0.5 for conventional hydrocarbon fuels. This effect leads to the

observation that flame height should more or less be constant with increasing pressure.

The effect of pressure on flame radius was investigated by measuring the radius of the flames

shown in Figure 4.8 in an effort to quantify the exponent on the pressure. As Figure 4.8

demonstrates, the computed flame radii for ethanol do not agree the expected theoretical be-

haviour for hydrocarbon but instead appear to be more proportional to p−0.35. The overall

slope is lower than expected but this can probably be attributed to the presence of hydroxide

in ethanol. Further investigation is required to fully assess this influence on the laminar co-flow

flame structure.


Pressure (atm)

Fla

me

Rad

ius

(mm

)

2 4 6 8 10 12 14 1618

0.6

0.8

1

1.2

1.4

1.6

1.8

2

DryerTheory

slope = -0.35

Figure 4.8: Flame radius as a function of pressure for ethanol using Dryer mechanism.

4.9 Concentrations of Intermediate Species

4.9.1 Acetylene Mass Fraction

The semi-empirical soot model employed in this work assumes acetylene as the sole precursor

to soot formation. Thus, this species is important in determining soot yield of ethanol. Since

the amount of acetylene present is an indicator of the soot yield, the acetylene mass fraction

contour is shown only for pressures which yields soot as seen in Figure 4.9. In general, both

mechanisms predicted the same amount of acetylene with amount of acetylene decreasing with

increasing pressure.

4.9.2 Carbon Monoxide Mass Fraction

Carbon monoxide (CO) is one of the products from the chemical reaction of acetylene. In

general, both mechanisms predicted the same amount of CO as shown in Figure 4.10.


Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_C2H2

0.0220.01950.0170.01450.0120.00950.0070.00450.002

DryerReduced Dryer

0.02370.0233

5 atm

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_C2H2

0.0220.01950.0170.01450.0120.00950.0070.00450.002

DryerReduced Dryer

10 atm

0.0224 0.0229

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_C2H2

0.0220.01950.0170.01450.0120.00950.0070.00450.002

DryerReduced Dryer

15 atm

0.02170.0212

Figure 4.9: Effect of reaction mechanism and pressure on acetylene mass fraction for ethanol

at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the bottom right

and left corners.


Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_CO

0.0350.031250.02750.023750.020.016250.01250.008750.005

DryerReduced Dryer

0.03880.0367

5 atm

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_CO

0.0350.031250.02750.023750.020.016250.01250.008750.005

DryerReduced Dryer

10 atm

0.0356 0.0383

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

y_CO

0.0350.031250.02750.023750.020.016250.01250.008750.005

DryerReduced Dryer

15 atm

0.03810.0349

Figure 4.10: Effect of reaction mechanism and pressure on carbon monoxide mass fraction for

ethanol at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the bottom

right and left corners.


4.10 Predicted Soot Formation

4.10.1 Contours of Soot Volume Fraction

For the case of 1 atm, the maximum soot volume fraction was determined to 0.0009 ppm

which is below what is measurable with the current experimental setup. Thus soot volume

fraction contours is not included for 1 atm. From Figure 4.11, the structure for predicted soot

distribution varied with increasing pressure. The soot volume fraction had an annular structure

and maximum concentrations occurred in the annular region. The annular structure was more

pronounced and thinner for increasing pressures.

4.10.2 Radial Profiles of Soot Concentration

Three axial locations have been chosen to compute the radial soot profiles as seen in Figure 4.12

for the Dryer mechanism. The three locations are the same the radial locations as mentioned

previously in Section 4.6.2. In general, the peak soot volume fraction moves towards the

centreline and soot yield increases as the pressure is increased. These similar trends are observed

for gaseous fuels as well.


Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4

-4

-2

0

2

4

6

8

10

12ppm

54.543.532.521.510.5

DryerReduced Dryer

5.48 ppm4.27 ppm

5 atm

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

ppm

2623.621.218.816.41411.69.26.84.42

DryerReduced Dryer

10 atm

27.26 ppm23.31 ppm

Radius (mm)

Hei

gh

t (m

m)

-4 -2 0 2 4-5

0

5

10

ppm

45403530252015105

DryerReduced Dryer

15 atm

47.44 ppm41.79 ppm

Figure 4.11: Effect of reaction mechanism and pressure on soot volume fraction for ethanol at

pressures of 5, 10 and 15 atmospheres. Peak soot volume fraction is indicated on the bottom

right and left corners.


Radius (mm)

So

ot

Vo

lum

e F

ract

ion

(p

pm

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

0.5 mm1.0 mm1.5 mm

5 atm

Radius (mm)

So

ot

Vo

lum

e F

ract

ion

(p

pm

)

0.0 0.5 1.0 1.50

5

10

15

20

250.5 mm1.0 mm1.5 mm

10 atm

Radius (mm)

So

ot

Vo

lum

e F

ract

ion

(p

pm

)

0 0.5 1 1.50

5

10

15

20

25

30

35

40

45

50

0.5 mm1.0 mm1.5 mm

15 atm

Figure 4.12: Radial soot volume fraction at three axial locations for ethanol for pressures of 5,

10 and 15 atm using Dryer mechanism.

4.10.3 Sooting Propensity

For a better measure of ethanol’s sooting propensity and its sensitivity to pressure, the variation

of the carbon conversion factor as a function of pressure was calculated rather than relying on

soot concentration alone. Previously, this parameter has been used by other researchers [77] to

quantify the effects of pressure on soot formation. The carbon conversion factor, ηs, is defined

as ms/mc where mc is defined as the carbon mass flow rate at the exit of the burner nozzle [78].


Pressure (atm)

Max

Fu

el C

arb

on

Co

nve

rsio

n t

o S

oo

t (%

)

4 6 8 10 12 14 16 18

0.005

0.01

0.015

0.02

Dryer

Slope = 2.3

Slope = 0.25

Figure 4.13: Effect of pressure on the maximum carbon conversion factor for ethanol using

Dryer mechanism.

The mass flux of soot through a horizontal cross-section is

ms = 2πρs

∫fvvr dr, (4.2)

where ρs = 1.9 g/cm3 is the density of soot [75], fv is the soot volume fraction and v is the

axial velocity. In calculating mc for ethanol, N2 was included since it is considered inert. As a

results, all the ethanol flames have the same carbon mass flow rate of 0.292 mg/s at the nozzle

exit. Referring to Figure 4.13, the maximum carbon conversion factor increased with pressure

but the exponent with pressure decreases as indicated by the value of slopes. Comparing with

the previous numerical study by Charest et al. [13] for methane, a pressure exponent of 2 was

found by Charest et al. for pressures in the range from 5 atm to 10 atm and a value of 0.5

was observed for the exponent in the range from 10 atm to 15 atm. In contrast, the results

depicted in Figure 4.13 indicate values for the pressure exponent of 2.3 and 0.25 respectively

for the same pressure ranges.

Chapter 5

Conclusions and Future Research

5.1 Conclusions I: One-Dimensional Laminar

Premixed Flames

In this work, one-dimensional laminar premixed flames of ethanol were simulated using the

Cantera software package. The converged solutions were then used to quantify the laminar

flame speed and adiabatic flame temperature as a function of equivalence ratio and pressure.

The numerical results were obtained using two different mechanisms. In general, the flame speed

calculated using both the mechanisms were within 10% of each other. The difference in adiabatic

flame temperature values between the two mechanisms were minimal. Since the difference

between the values obtained by the two mechanisms of ethanol are small, the reduced Dryer

mechanism can used to calculate various fuel properties of interest relatively quickly without

significant errors. This mechanism is expected to be particularly useful in more complicated

simulations.

The laminar flame speed and adiabatic flame temperature as a function of stoichiometry and

pressure was evaluated. In general, the flame speed increased with increasing equivalence ratio

and decreased after a particular equivalence ratio (slightly fuel rich conditions). While as the

pressure increases, the flame speed was found to decrease quite dramatically. When the pressure

was increased from 1 to 25 atmospheres, the flame speed generally decreased by at least 70%.

Stoichiometry effects were common for all the fuels in that the peak adiabatic flame temperature

was attained at an equivalence ratio of approximately 1.1 at 1 atmosphere pressure. This was

attributed to the occurrence of dissociation reactions that in effect that take away heat from the

reactions. This effect was noticeably smaller at higher pressures and flame temperature peaks

56

Chapter 5. Conclusions and Future Research 57

were found to be shift much closer to or at stoichiometric point at a pressure of 25 atmospheres.

The proportionality of the flame speed with p−0.5 was also investigated. The numerical results

were found to be in rather good agreement with this theoretical expectation.

5.2 Conclusions II: Axisymmetric Laminar Co-Flow

Diffusion Flames

Two-dimensional laminar axisymmetric co-flow diffusion flames were also simulated for ethanol

utilizing two different reaction mechanisms to assess the influence of pressure and chemical ki-

netic mechanism on flame structure and soot yield. For constant mass fuel flows, the predicted

flame heights essentially remained constant. The differences in stoichiometric predicted lami-

nar co-flow diffusion flame structure and heights obtained using the Dryer and reduced Dryer

chemical kinetic mechanisms were noted to be negligible. This is expected since the reduced

Dryer mechanism was derived from the full Dryer mechanism.

The effect on flame radius of pressure was also investigated for ethanol using the Dryer mecha-

nism. It was observed that the predicted flame radius from numerical simulations was propor-

tional to p−0.35. However, theory for conventional hydrocarbon fuels predicts p−0.5 [37]. This

difference was attributed to the presence of hydroxide which is a diatomic anion.

The influence of pressure on soot formation have similiar observation to other gaseous fuels [11,

12,36], i.e, soot yield increases with pressure. The maximum carbon conversion factor increased

with pressure but the exponent with pressure decreases as was shown in Figure 4.13 of the

previous chapter. This could be attributed to the presence of nitrogen dilution.

5.3 Recommendations for Future Research

The present numerical study of the combustion characteristics of ethanol includes some of the

first numerical research on the sooting propensity of this liquid fuel at elevated pressures. How-

ever, full experimental comparisons of soot volume fraction and temperature fields are required

to fully assess the numerical results for ethanol missing from this study. This research should

be extended by investigating other ethanol mechanisms such as Saxena [15] and Marinov [79]

for numerical simulations. There are several other attractive liquid fuels and biofuels such as

n-heptane, biodiesel, etc. Currently, the combustion laboratory at UTIAS has conducted ex-

periments for n-heptane which has a more complicated fuel structure as compared to ethanol.

Chapter 5. Conclusions and Future Research 58

Future work should include numerical investigation of the fuel-n-heptane which is also a suitable

surrogate for biodiesel [80] using the current numerical framework.

References

[1] “2010 World Energy Outlook,” 2010.

[2] Jason Y.W. Lai, Kuang C.Lin, A. V., “Biodiesel Combustion,” Process in Energy and

Combustion Science, 2011.

[3] Nigam, P. and Singh, A., Progress in Energy and Combustion Science, Vol. 377, 2011,

pp. 52–68.

[4] Cavaliere, A. and Ragucci, R., “Gaseous diffusion flames: simple structures and their

interaction,” Progress in Energy and Combustion Science, Vol. 275, 2001, pp. 47–85.

[5] Karcher, B., Mohler, O., Demott, P., Pechtl, S., and Yu, F., “Insights into the role of soot

aerosols in cirrus cloud formation,” Atmospheric Chemistry and Physics, Vol. 7, 2007,

pp. 42034227.

[6] Charest, M. R. J., Numerical Modelling of Sooting Laminar Diffusion Flames at Elevated

Pressures and Microgravity , Ph.D. thesis, University of Toronto, Toronto, ON, 2010.

[7] Joo, P. H., Charest, M. R. J., Groth, C. P. T., and Gulder, O. L., “Comparison of structures

of laminar methane-oxygen and methane-air diffusion flames from atmospheric to 60 atm,”

Combustion and Flame, Vol. 160, 2013, pp. 1990–1998.

[8] Barua, A., Soot Formation in Diffusion Flames of Alternative Turbine Fuels at Elevated

Pressures, Master’s thesis, University of Toronto, Toronto, ON, 2012.

[9] Charest, M. R. J., Joo, H. I., Gulder, O. L., and Groth, C. P. T., “Experimental and

numerical study of soot formation in laminar ethylene diffusion flames at elevated pressures

from 10 to 35 atm,” Proceedings of the Combustion Institute, Vol. 33, 2011, pp. 549–557.

[10] Karatas, A. E., Intasopa, G., and Gulder, O. L., “Sooting behaviour of n-heptane laminar

diffusion flames at high pressures,” Combustion and Flame, Vol. 160, 2013, pp. 1650–1656.

59

References 60

[11] Charest, M. R. J., Groth, C. P. T., and Gulder, O. L., “A computational framework for

predicting laminar reactive flows with soot formation,” Combustion Theory and Modelling ,

Vol. 14, 2010, pp. 793–825.

[12] Charest, M. R. J., Groth, C. P. T., and Gulder, O. L., “A numerical study on effects of

pressure and gravity in laminar ethylene diffusion flames,” Combustion and Flame, 2011.

[13] Charest, M. R. J., Gulder, O. L., and Groth, C. P. T., “Numerical and experimental study

of soot formation in laminar diffusion flames burning simulated biogas fuels at elevated

pressures,” Combustion and Flame, Vol. 161, 2014, pp. 2678–2691.

[14] Karatas, A. E., High-Pressure Soot Formation and Diffusion Flame Extinction Character-

istics of Gaseous and Liquid Fuels, Ph.D. thesis, University of Toronto, Toronto,Ontario,

2014.

[15] Priyank Saxena, F. A., “Numerical and experimental studies of ethanol flames,” Proceed-

ings of the Combustion Institute, 2007.

[16] M.Haas, F., Chaos, M., and L.Dryer, F., “Low and intermediate temperature oxidation of

ethanol and ethanol-PRF blends: An Experimental and modeling study,” Combustion and

Flame, 2009.

[17] Mohanty, S., Behera, S., Swain, M., and Ray, R., “Bioethanol production from mahula

(Madhuca latifolia L.) flowers by solid-state fermentation,” Applied Energy , Vol. 86, 2009,

pp. 640–644.

[18] Pimentel, D., Biofuels, solar and wind as renewable energy systems: benefits and risks,

Springer-Verlag, Dordrecht (Netherlands), 2008.

[19] Smith, A., “Prospects for increasing starch and sucrose yields for bioethanol production,”

The Plant Journal , Vol. 54, 2008, pp. 546–558.

[20] Enguidanos, M., Soria, A., Kavalov, B., and Jensen, P., “Techno-economic analysis of

bio-alcohol production in the EU: a short summary for decision-makers,” 2002.

[21] Haynes, B. S. and Wagner, H. G., “Soot Formation,” Progress in Energy and Combustion

Science, Vol. 7, 1981, pp. 229–273.

[22] Guo, H., Liu, F., Smallwood, G. J., and Gulder, O. L., “Numerical modeling of a two

dimensional axisymmetric laminar ethylene-air diffusion flame,” Proceeeding of ASME ,

Vol. 1, Ananheim, CA, United States, 2001, pp. 869–877.

References 61

[23] Fairweather, M., Jones, W. P., and Lindstedt, R. P., “Predictions of radiative transfer from

a turbulent reacting jet in a cross-wind,” Combustion and Flame, Vol. 89, 1992, pp. 45–63.

[24] Lindstedt, R. P., “A simplified mechanism for soot formation in non-premixed flames,”

Aerothermodynamics in Combustors, 1992, pp. 145–156.

[25] Fairweather, M., Jones, W. P., Ledin, H. S., and Lindstedt, R. P., “Predictions of soot for-

mation in turbulent, non-premixed propane flames,” Proceedings of Combustion Institute,

Vol. 24, 1992, pp. 1067–1074.

[26] Joo, H. I. and Gulder, O. L., “Soot formation and temperature field structure in co-flow

laminar methane-air diffusion flames at pressures from 10 to 60 atm,” Proceedings of the

Combustion Institute, Vol. 32, 2009, pp. 769–775.

[27] Barua, A., Soot formation in diffusion flames of alternative turbine fuels at elevated pres-

sures, Ph.D. thesis, University of Toronto, Toronto, ON, 2011.

[28] Mandatori, P. M. and Gulder, O. L., “Soot formation in laminar ethane diffusion flames

at pressures from 0.2 to 3.3 MPa,” Proceedings of the Combustion Institute, Vol. 33, 2010,

pp. 577–584.

[29] Turns, S. R., An Introduction to Combustion, McGraw-Hill International, Singapore, 2006.

[30] Law, C. K., Combustion Physics, Cambridge University Press, New York, 2006.

[31] Glassman, I., “Soot formation in combustion processes,” Proceedings of the Combustion

Institute, Vol. 22, 1988, pp. 295–311.

[32] Miller, I. M. and Maahs, H. G., “High-Pressure Flame Systems for Pollution Studies with

Results for Methane-Air Diffusion Flames,” TN D-8407, NASA, 1977.

[33] McCrain, L. L. and Roberts, W. L., “Measurements of the soot volume field in laminar

diffusion flames at elevated pressures,” Combustion and Flame, Vol. 140, 2005, pp. 60–69.

[34] Roper, F. G., “The prediction of laminar jet diffusion flame sizes: Part I. theoretical

model,” Combustion and Flame, Vol. 29, 1977, pp. 219–226.

[35] Roper, F. G., Smith, C., and Cunningham, A. C., “The prediction of laminar jet diffusion

flame sizes: Part II. experimental verification,” Combustion and Flame, Vol. 29, 1977,

pp. 227–234.

[36] Liu, F., Thomson, K. A., Guo, H., and Smallwood, G. J., “Numerical and experimental

study of an axisymmetric coflow laminar methaneair diffusion flame at pressures between

5 and 40 atmospheres,” Combustion and Flame, Vol. 146, 2006, pp. 456–471.

References 62

[37] McRain, L. and Roberts, W., “Measurements of the soot volume field in laminar diffusion

flames at elevated pressures,” Combustion and Flame, Vol. 140, 2005, pp. 60–69.

[38] Thomson, K., Gulder, O. L., Weckman, E., Fraser, R., Smallwood, G., and Snelling, D.,

“Soot concentration and temperature measurements in co-annular, nonpremixed CH4/air

laminar flames at pressures up to 4 MPa,” Combustion and Flame, Vol. 140, 2005, pp. 222–

232.

[39] Bento, D. S., Thomson, K. A., and Gulder, O. L., “Soot formation and temperature field

structure in laminar propane-air diffusion flames at elevated pressures,” Combustion and

Flame, Vol. 146, 2006, pp. 765–778.

[40] California Institute of Technology, “CANTERA Release 1.8,”

http://code.google.com/p/cantera/, 2009.

[41] Monteiro, E., Bellenoue, M., Sotton, J., Moreira, N. A., and Malheiro, S., “Laminar burn-

ing velocities and Markstein numbers of syngasair mixtures,” Fuel , Vol. 89, 2009, pp. 1985–

1991.

[42] Natarajan, J., Lieuwen, T., and Seitzman, J., “Laminar ame speeds of H2/CO mix-

tures:Effect of CO2 dilution, preheat temperature, and pressure,” Combustion and Flame,

Vol. 151, 2007, pp. 104–119.

[43] de Vries, J., Lowry, W. B., Serinyel, Z., Curran, H. J., and Petersen, E. L., “Laminar flame

speed measurements of dimethyl ether in air at pressures up to 10 atm,” Fuel , Vol. 90,

2011, pp. 331–338.

[44] Kumgeh, B. A., Shock Tube Studies and Chemical Kinetic Modeling of Oxygenated Hydro-

carbon Ignition, Ph.D. thesis, McGill University, Montreal,Quebec, 2011.

[45] Charest, M. R. J., Groth, C. P. T., and Gulder, O. L., “Solution of the equation of

radiative transfer using a Newton-Krylov approach and adaptive mesh refinement,” Journal

Of Computational Physics, 2010.

[46] Northrup, S. and Groth, C., “Solution of laminar diffusion flames using a parallel adaptive

mesh refinement algorithm,” 43rd AIAA Aerospace Sciences Meeting and Exhibition, 2005,

pp. AIAA paper 2005–0547.

[47] Weiss, J. and Smith, W., “Preconditioning Applied to Variable and Constant Density

Flows,” AIAA Journal , Vol. 33, 1995, pp. 2050–2057.

References 63

[48] Carlson, B. and Lathrop, K., “Transport theory - the method of discrete ordinates,” Com-

puting Methods in Reactor Physics, 1968, pp. 171–266.

[49] Liu, F., Smallwood, G., and Gulder, O., “Application of the statistical narrow-band

correlated-k method to low-resolution spectral intensity and radiative heat transfer cal-

culations - effects of the quadrature scheme,” International Journal of Heat and Mass

Transfer , Vol. 43, 2000, pp. 3119–3135.

[50] Lacis, A. and Oinas, V., “A description of the correlated k distribution method for mod-

eling nongray gaseous absorption, thermal emission, and multiple scattering in vertically

inhomogeneous atmospheres,” Journal of Geophysical Research, Vol. 96, 1991, pp. 9027–

9063.

[51] Fairweather, M., Jones, W., , and Lindstedt, R., “Predictions of radiative transfer from a

turbulent reacting jet in a cross-wind,” Combustion and Flame, Vol. 89, 1992, pp. 45–63.

[52] Leung, K., Lindstedt, R., and Jones, W., “A simplified reaction mechanism for soot for-

mation in nonpremixed flames,” Combustion and Flame, Vol. 79, 1991, pp. 289–305.

[53] Fletcher, T. H. and Goodwin, D. G., “Cantera introduction workshop,”

http://www.et.byu.edu/ ˜tom/classes/641/Cantera/Workshop/flames.pdf.

[54] Northrup, S. and Groth, C., “Parallel implicit adaptive mesh refinement algorithm for

solution of laminar combusting flows,” Proceedings of the 14th Annual Conference of the

CFD Society of Canada, Kingston, ON, Canada, 16-18 July 2006, pp. 221–228.

[55] Gao, X. and Groth, C., “Parallel adaptive mesh refinement scheme for turbulent non-

premixed combusting flow prediction,” 44th AIAA Aerospace Sciences Meeting and Ex-

hibit , Reno, Nevada, USA, 9-12 January 2006, pp. 17450–17463.

[56] Leer, B. V., “Towards the ultimate conservative difference scheme. V. A second-order sequel

to Godunov’s method,” Journal of Computational Physics, Vol. 32, 1979, pp. 101–136.

[57] Coirier, W. and Powell, K., “Solution-adaptive Cartesian cell approach for viscous and

inviscd flows,” AIAA Journal , Vol. 34, 1996, pp. 938–945.

[58] Carlson, B. and Lathrop, K., “Transport theory - the method of discrete ordinates,” Com-

puting Methods in Reactor Physics, 1968, pp. 171266.

[59] Ventakeshwaran, S. and Merkle, C. L., “Analysis of Preconditioning Methods for the Euler

and Navier-Stokes Equations,” Tech. rep., University of Tennessee Space Institute, 1999.

References 64

[60] Choi, Y. and Merkle, C., “The application of preconditioning in viscous flows,” Journal of

Computational Physics, Vol. 105, 1993, pp. 207–223.

[61] Godunov, S., “A difference method for numerical calculation of discontinuous solutions of

the equations of hydrodynamics,” Mat. Sb., Vol. 47, 1959, pp. 271–306.

[62] Roe, P., “Approximate Riemann solvers, parameter vectors and difference schemes,” Jour-

nal of Computational Physics, Vol. 43, 1981, pp. 357–372.

[63] Harten, A., “High resolution schemes for hyperbolic conservation laws,” Journal of Com-

putational Physics, Vol. 49, 1983, pp. 357–393.

[64] Shuen, J., “Inviscid flux-splitting algorithms for real gases with non-equilibrium chemistry,”

Journal of Computational Physics, Vol. 90, 1990, pp. 370–395.

[65] Barth, T. and Jespersen, D., “The design and application of upwind schemes on unstruc-

tured meshes,” AIAA Journal , Vol. 89, 1989, pp. 1–12.

[66] Zingg, D. and Yarrow, M., “A method of smooth bivariate interpolation for data given

on a generalized curvilinear grid,” SIAM Journal on Scientific Computing , Vol. 13, 1992,

pp. 687–693.

[67] Northrup, S. and Groth, C., “Parallel Implicit Adaptive Mesh Renement Scheme for Body-

Fitted Multi-Block Mesh,” Proceedings of the 17th AIAA CFD conference, Toronto, ON,

Canada, 6-9 June 2005.

[68] Saad, Y. and Schultz, M., “GMRES: A generalized minimal residual algorithm for solv-

ing nonsymmetric linear systems,” SIAM Journal on Scientific Computing , Vol. 7, 1986,

pp. 856–859.

[69] Saad, Y., “Krylov subspace methods on supercomputers,” SIAM Journal on Scientific

Computing , Vol. 10, 1989, pp. 1200–1232.

[70] Knoll, D. and Keyes, D., “Jacobian-free Newton-Krylov methods: A survey of approaches

and applications,” Journal of Computational Physics, Vol. 193, 2004, pp. 357–397.

[71] Modest, M. F., Radiative Heat Transfer , Academic Press, New York, 2nd ed., 2003.

[72] Bradley, D., Lawes, M., and Mansour, M., “Explosion bomb measurements of ethanolair

laminar gaseous flame characteristics at pressures up to 1.4MPa,” Combustion and Flame,

Vol. 156, 2009, pp. 1462–1470.

References 65

[73] Lu, T. and Law, C. K., “A directed relation graph method for mechanism reduction,”

Proceedings of the Combustion Institute, Vol. 30, 2005, pp. 1333–1341.

[74] Okajima, S., Iinuma, K., Yamaguchi, S., and Kumagai, S., “Measurement of slow burn-

ing velocities and their pressure dependence using a zero-gravity method,” Symposium

(International) on Combustion, Vol. 20, 1985, pp. 1951–1956.

[75] Guo, H., Liu, F., Smallwood, G. J., and Gulder, O. L., “The flame preheating effect on

numerical modelling of soot formation in a two-dimensional laminar ethylene-air diffusion

flame.” Combustion Theory and Modelling , Vol. 6, 2002, pp. 173–187.

[76] Skeen, S., Yablonsky, G., and Axelbaum, R., “Characteristics of non-premixed oxygen-

enhanced combustion: I. The presence of appreciable oxygen at the location of maximum

temperatures,” Combustion and Flame, Vol. 156, 2009, pp. 2145–2152.

[77] Rudinger, G., Fundamentals of Gas-Particle Flow , Elsevier Scientific, 1980.

[78] Flower, W. L. and Bowman, C. T., “Soot production in axisymmetric laminar diffusion

flames at pressures from one to ten atmospheres,” Proceeding of Combustion Institute,

Vol. 21, 1988, pp. 1115–1124.

[79] Marinov, N. M., “A detailed chemical kinetic model for high temperature ethanol oxida-

tion,” International Journal of Chemical Kinetics, Vol. 31, No. 3, 1999, pp. 183–220.

[80] Liu, W., Sivaramakrishnan, R., Davis, M. J., Som, S., and Longman, D. E., “Develop-

ment of a Reduced Biodiesel Surrogate Model for Compression Ignition Engine Modeling,”

Proceeding of Combustion Institute, Vol. 160, 2013, pp. 2712–2728.

numerical investigation of sooting propensity of ethanol ... · numerical investigation of sooting...

Documents